Find the area of the finite region enclosed between the following curves:

y= 0.3x + 5 sin x and y = 2x

Thank you for helping!!

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- Aug 16th 2008, 10:17 PMTangeraIntergration
Find the area of the finite region enclosed between the following curves:

y= 0.3x + 5 sin x and y = 2x

Thank you for helping!! - Aug 16th 2008, 10:33 PMo_O
It helps if you have a visual of what is going on. First we find the intersection points to determine the limits of integration:

$\displaystyle 0.3x + 5\sin x = 2x \: \: \iff \: \: \sin x = 0.34 x$

This is impossible to solve algebraically so you're going have to obtain the intersection points via your graphing calculator. Let's call these intersection points x = a, 0, and b.

If you graph it out, you'll see y = 2x is greater than 0.3x + 5sin x over the interval $\displaystyle [a,0]$ and the other way around for the interval $\displaystyle [0,b]$. (Remember, area enclosed by two curves is given by the integral of the upper minus the lower function) So,

$\displaystyle A = \int_{a}^{0} \left[2x - \left(0.3x + 5\sin x\right)\right] dx + \int_{0}^{b} \left[(0.3x + 5\sin x) - 2x\right]dx$